semantic proof (idea)

A form of proof in propositional logic that determines whether a proposition follows from other propositions, by working through the truth tables for the premisses, and checking whether the consequent is true whenever the premisses are. If so, the consequent is proved. The method is guaranteed to be correct but is impossibly cumbersome beyond small examples because the size of the calculation is exponential on the number of premisses.

Contrast syntactic proof, which uses logical operations such as modus ponens to rearrange and combine theorems derived from the premisses, but which is not guaranteed to find an elegant solution.

Suppose we know that (A) I won't need my umbrella just in case it doesn't rain today, (B) it always rains on Saturdays, and (C) today is Saturday. We want to prove whether I'll need my umbrella. Let these propositions be built out of simple ones: